Epistemic game theory∗ Eddie Dekel and Marciano Siniscalchi Tel Aviv and Northwestern University, and Northwestern University January 14, 2014 Contents 1 Introduction and Motivation2 1.1 Philosophy/Methodology.................................4 2 Main ingredients5 2.1 Notation...........................................5 2.2 Strategic-form games....................................5 2.3 Belief hierarchies......................................6 2.4 Type structures.......................................7 2.5 Rationality and belief...................................9 2.6 Discussion.......................................... 10 2.6.1 State dependence and non-expected utility................... 10 2.6.2 Elicitation...................................... 11 2.6.3 Introspective beliefs................................ 11 2.6.4 Semantic/syntactic models............................ 11 3 Strategic games of complete information 12 3.1 Rationality and Common Belief in Rationality..................... 12 3.2 Discussion.......................................... 14 3.3 ∆-rationalizability..................................... 14 4 Equilibrium Concepts 15 4.1 Introduction......................................... 15 4.2 Subjective Correlated Equilibrium............................ 16 4.3 Objective correlated equilibrium............................. 17 4.4 Nash Equilibrium...................................... 21 4.5 The book-of-play assumption............................... 22 4.6 Discussion.......................................... 23 4.6.1 Condition AI.................................... 23 4.6.2 Comparison with Aumann(1987)......................... 24 4.6.3 Nash equilibrium.................................. 24 ∗We thank Drew Fudenberg for his detailed comments, and Robert Molony and Luciano Pomatto for excellent research assistantship. We also thank Pierpaolo Battigalli, Adam Brandenburger, Yi-Chun Chen, Amanda Frieden- berg, Joe Halpern, Qingmin Liu, Andres Perea, two anonymous referees, and the editors, Peyton Young and Shmuel Zamir, for helpful feedback. Eddie Dekel gratefully acknowledges financial support from NSF grant SES-1227434. 1 5 Strategic-form refinements 25 6 Incomplete information 28 6.1 Introduction......................................... 28 6.2 Interim Correlated Rationalizability........................... 30 6.3 Interim Independent Rationalizability.......................... 32 6.4 Equilibrium concepts.................................... 35 6.5 ∆-rationalizability..................................... 36 6.6 Discussion.......................................... 37 7 Extensive-form games 38 7.1 Introduction......................................... 38 7.2 Basic ingredients...................................... 40 7.3 Initial CBR......................................... 43 7.4 Forward Induction..................................... 45 7.4.1 Strong Belief.................................... 45 7.4.2 Examples...................................... 46 7.4.3 RCSBR and Extensive-Form Rationalizability.................. 47 7.4.4 Discussion...................................... 50 7.5 Backward Induction.................................... 52 7.6 Equilibrium......................................... 52 7.7 Discussion.......................................... 54 Strategies...................................... 54 Incomplete information.............................. 55 8 Admissibility 55 8.1 Basics............................................ 56 8.2 Assumption and Mutual Assumption of Rationality................... 57 8.3 Characterization...................................... 57 8.4 Discussion.......................................... 59 8.4.1 Issues in the characterization of IA........................ 59 Relationship with Brandenburger, Friedenberg, and Keisler(2008)...... 59 Full-support beliefs................................. 59 Common vs. mutual assumption of admissibility................ 60 8.4.2 Extensive-form analysis and strategic-form refinements............. 60 LPSs and CPSs................................... 60 Strong belief and assumption........................... 61 Admissibility and sequential rationality..................... 61 1 Introduction and Motivation Epistemic game theory formalizes assumptions about rationality and mutual beliefs in a formal language, then studies their behavioral implications in games. Specifically, it asks: what do different notions of rationality and different assumptions about what players believe about ... what others believe about the rationality of players imply regarding play in a game? A well-known example is the equivalence between common belief in rationality and iterated deletion of dominated strategies. The reason why it is important to be formal and explicit is the standard one in economics. Solution concepts are often motivated intuitively in terms of players' beliefs and their rationality. 2 However, the epistemic analysis may show limitations in these intuitions, reveal what additional assumptions are hidden in the informal arguments, clarify the concepts or show how the intuitions can be generalized. We now consider a number of examples. Backwards induction was long thought to be obviously implied by \common knowledge of rationality." The epistemic analysis showed flaws in this intuition and it is now understood that the characterization is much more subtle (Sections 7.4.3 and 7.5). Next, consider the solution concept that deletes one round of weakly dominated strategies and then iteratively deletes strictly dominated strategies. This concept was first proposed because it is robust to payoff perturbations, which were interpreted as a way to perturb players' rationality. Subsequent epistemic analysis showed this concept is exactly equivalent to \almost common belief" of rationality and of full-support conjectures { an explicit robustness check of common belief in rationality (see Section5). Thus the epistemic analysis generalizes and formalizes the connection of this concept to robustness. The common-prior assumption (Section 4.3) is used to characterize Nash equilibrium with n > 2 players, but not needed for two-player games (compare Theorems5 and7). This result highlights the difference between the concept in these environments. Furthermore, the common-prior is known to be equivalent to no betting when uncertainty is exogenous. We argue that the interpretation of the common-prior assumption and its connection to no-betting results must be modified when uncertainty is endogenous, e.g, about players' strategies (see Example4). Finally, recent work has shown how forward induction and iterated deletion of weakly domi- nated strategies can be characterized. These results turn out to identify important, non-obvious, assumptions and require new notions of \beliefs." Moreover, they clarify the connection between these concepts. (See Section 7.4.4.) Epistemic game theory may also help provide a rationale, or ‘justification,’ for or against specific solution concepts. For instance, in Section6 we identify those cases where interim independent rationalizability is and is not a \suitable" solution concept for games of incomplete information. We view non-epistemic justifications for solution concepts as complementary to the epistemic approach. For some solution concepts, such as forward induction, we think the epistemic analysis is more insightful. For others, such as Nash equilibrium, learning theory may provide the more compelling justification. Indeed, we do not find the epistemic analysis of objective equilibrium notions (Section4) entirely satisfactory. This is because the epistemic assumptions needed are often very strong and hard to view as a justification of a solution concept. Moreover, except for special cases (e.g. pure-strategy Nash equilibrium), it is not really possible to provide necessary and sufficient epistemic conditions for equilibrium behavior (unless we take the view that mixed strategies are actually available to the players). Rather, the analysis constitutes a fleshing-out of the textbook interpretation of equilibrium as `rationality plus correct beliefs.' To us this suggests that equilibrium behavior cannot arise out of strategic reasoning alone. Thus, as discussed above, this epistemic analysis serves the role of identifying where alternative approaches are required to justify standard concepts. While most of the results we present are known from the literature, we sometimes present them differently, to emphasize how they fit within our particular view. We have tried to present a wide swath of the epistemic literature, analyzing simultaneous-move games as well as dynamic games, considering complete and incomplete information games, and exploring both equilibrium and non- equilibrium approaches. That said, our choice of specific topics and results is still quite selective and we admit that our selection is driven by the desire to demonstrate our approach (discussed next), as well as our interests and tastes. Several insightful and important papers could not be included because they did not fit within our narrative. More generally, we have ignored several literatures. The connection with the robustness literature mentioned above (see Kajii and Morris 3 (1997b) for a survey) is not developed. Nor do we study self-confirming based solution concepts (Fudenberg and Levine, 1993; Battigalli, 1987; Rubinstein and Wolinsky, 1994).1 Moreover, we do not discuss epistemics and k-level thinking (Crawford, Costa-Gomes, and Iriberri, 2012; Kets, 2012) or unawareness (see Schipper, 2013, for a comprehensive